Centers of Triangles or Points of Concurrency

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# Centers of Triangles or Points of Concurrency - PowerPoint PPT Presentation

Centers of Triangles or Points of Concurrency. Geometry October 28, 2013. OBJECTIVE. You will learn how to construct perpendicular bisectors, angle bisectors, medians and altitudes of triangles constructed. Today’s Agenda. Triangle Segments Median Altitude Perpendicular Bisector

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### Centers of Triangles or Points of Concurrency

Geometry

October 28, 2013

OBJECTIVE

You will learn how to construct perpendicular bisectors, angle bisectors, medians and altitudes of triangles constructed

Today’s Agenda
• Triangle Segments
• Median
• Altitude
• Perpendicular Bisector
• Angle Bisector
• Triangle Centers
• Centroid
• Orthocenter
• Circumcenter
• Incenter
VOCABULARY
• A median of a triangle is a segment that connects a vertex to the midpoint of the opposite side.
• The altitude of a triangle is the perpendicular distance from one of its bases to the opposite vertex. In other words, the altitude is a segment that is perpendicular to one side and reaches the point across from that side.
• A perpendicular bisector of a line segment isa) perpendicular to it, and b) bisects it.
• An angle bisector of a triangle is a segment that divides one of its angles into two congruent pieces. The segment connects to the opposite side

Medians

Median

vertex to midpoint

Example 1

M

D

P

C

What is NC if NP = 18?

MC bisects NP…so 18/2

9

N

If DP = 7.5, find MP.

15

7.5 + 7.5 =

How many medians does a triangle have?

Three – one from each vertex

The medians of a triangle are concurrent.

The intersection of the medians is called the CENTRIOD.

They meet in a single point.

Theorem

The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.

2x

x

Example 2

In ABC, AN, BP, and CM are medians.

If EM = 3, find EC.

C

EC = 2(3)

N

P

E

EC = 6

B

M

A

Example 3

In ABC, AN, BP, and CM are medians.

If EN = 12, find AN.

C

AE = 2(12)=24

AN = AE + EN

N

P

E

AN = 24 + 12

B

AN = 36

M

A

C

N

P

E

B

M

A

Example 4

In ABC, AN, BP, and CM are medians.

If EM = 3x + 4 and CE = 8x, what is x?

x = 4

C

N

P

E

B

M

A

Example 5

In ABC, AN, BP, and CM are medians.

If CM = 24 what is CE?

CE = 2/3CM

CE = 2/3(24)

CE = 16

Angle Bisector

Angle Bisector

vertex to side cutting angle in half

Example 1

W

X

1

2

Z

Y

Example 2

F

I

G

5(x – 1) = 4x + 1

5x – 5 = 4x + 1

x = 6

H

How many angle bisectors does a triangle have?

three

The angle bisectors of a triangle are ____________.

concurrent

The intersection of the angle bisectors is called the ________.

Incenter

Point P is called the __________.

Incenter

A

8

D

F

L

C

B

E

Example 4

LF, DL, EL

The angle bisectors of triangle ABC meet at point L.

• What segments are congruent?
• Find AL and FL.

Triangle ADL is a right triangle, so use Pythagorean thm

AL2 = 82 + 62

AL2 = 100

AL = 10

FL = 6

6

Altitude

Altitude

vertex to opposite side and perpendicular

The altitude is the “true height” of the triangle.

YES

NO

YES

How many altitudes does a triangle have?

Three

The altitudes of a triangle are concurrent.

The intersection of the altitudes is called the ORTHOCENTER.

Perpendicular Bisector

Perpendicular Bisector

midpoint and perpendicular

(don't care about no vertex)

Example 2: Find x

3x + 4

5x - 10

x = 7

How many perpendicular bisectors does a triangle have?

Three

The perpendicular bisectors of a triangle are concurrent.

The intersection of the perpendicular bisectors is called the CIRCUMCENTER.

PA = PB = PC

Find DA.

DA = 6

BA = 12

• Find BA.
• Find PC.

PC = 10

• Use the Pythagorean Theorem to find DP.

B

6

DP2 + 62 = 102

DP2 + 36 = 100

DP2 = 64

DP = 8

10

D

P

A

C

Tell if the red segment is an altitude, perpendicular bisector, both, or neither?

NEITHER

ALTITUDE

PER. BISECTOR

BOTH

Sum It Up

bisector

bisector

which is…

Figure

concurrent at..

circumcenter

equidistant from vertices

incenter

equidistant from sides

median

centroid

2/3 distance from vertices to midpoint

altitude

orthocenter

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